Estimation of location parameters for spherically symmetric distributions

Abstract

Estimation of the location parameters of a p × 1 random vector X with a spherically symmetric distribution is considered under quadratic loss. The conditions of Brandwein and Strawderman [Ann. Statist. 19(1991Statist. 19( ) 1639Statist. 19( -1650 under which estimators of the formwhere V has a uniform distribution in the sphere centered at with a radius R, and (iii) 0 < a 1/[pE 0 ( X −2 )]. In this paper, we not only drop their condition (ii) to show the dominance of X + ag(X) over X, but also obtain a new bound for a which is sometimes better than that obtained by Brandwein and Strawderman. Specifically, the new bound of a is 0 < a < [ 1 /(p 2 −1 )][1 − (p − 1) 1 /(p −1 2 )] −1 with i =E 0 ( X i ) for i =−1, 1, 2. The generalization to concave loss functions is also considered. Additionally, we investigate estimators of the location parameters when the scale is unknown and the observation contains a residual vector.

A note on classical Stein-type estimators in elliptically contoured models

“…Accordingly, Brandwein and Strawderman (1991), Ouassou and Strawderman (2002) and Xu and Izmirlian (2006) extended the similar results to that of Stein (1981) to the class of spherically symmetric distributions (SSDs). In their approaches they used the divergence theorem to prove improvement of the underlying estimators.…”

“…It would be interesting to extend our results to the more general case as considered in Brandwein and Strawderman (1991) and Xu and Izmirlian (2006), which leaves for further researches. …”